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Unformatted text preview: 3.1 Chapter Three Vector Functions 3.1 Relations and Functions We begin with a review of the idea of a function. Suppose A and B are sets. The Cartesian product A B × of these sets is the collection of all ordered pairs ( , ) a b such that a A ∈ and b B ∈ . A relation R is simply a subset of A B × . The domain of R is the set dom R = { :( , ) } a A a b R ∈ ∈ . In case A = B and the domain of R is all of A , we call R a relation on A . A relation R A B ⊂ × such that ( , ) a b R ∈ and ( , ) a c R ∈ only if b = c is called a function . In other words, if R is a function, and a dom R ∈ , there is exactly one ordered pair ( , ) a b R ∈ . The second “coordinate” b is thus uniquely determined by a . It is usually denoted R a ( ) . If R A B ⊂ × is a relation, the inverse of R is the relation R B A- ⊂ × 1 defined by R b a a b R- = ∈ 1 {( , ):( , ) } . Example Let A be the set of all people who have ever lived and let S A A ⊂ × be the relation defined by S a b b a = {( , ): } is the mother of . The S is a relation on A , and is, in fact, a function. The relation S- 1 is not a function, and domS A- ≠ 1 . The fact that f A B ⊂ × is a function with dom f = A is frequently indicated by writing f A B : → , and we say f is a function from A to B . Very often a function f is defined by specifying the domain, and giving a recipe for finding f(a) . Thus we may define the function f from the interval [0,1] to the real numbers by f x x ( ) = 2 . This says that f is the collection of all ordered pairs ( , ) x x 2 in which x ∈ [ , ] 01 . Exercises 3.2 1 . Let A be the set of all Georgia Tech students, and let B be the set of real numbers. Define the relation W A B ⊂ × by W a b b a = {( , ): } is the weight (in pounds) of . Is W a function? Is W- 1 a function? Explain. 2 . Let X be set of all states of the U. S., and let Y be the set of all U. S. municipalities. Define the relation c X Y ⊂ × by c x y y x = {( , ): } is the capital of . Explain why c is a function, and find c (Nevada), c (Missouri), and c (Kentucky). 3 . With X , Y as in Exercise 2 , let b be the function b x y y x = {( , ): } is the largest city in . a)What is b (South Carolina)? b)What is b (California)? c)Let f c b = ∩ , where c is the function defined in Exercise 2 . Find dom f ....
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